Question

Solve the equation \(15-3 x=3(x-7)+11\).

Short answer

Answer: x = 25/6

Step-by-step solution

Distribute and combine like terms

First, we will distribute the 3 on the right side of the equation by multiplying it with each term inside the parentheses: $$15 - 3x = 3(x - 7) + 11 \Rightarrow 15 - 3x = 3x - 21 + 11$$ Next, we will combine like terms on both sides. $$15 -3x = 3x - 10$$

Move terms with the variable to one side and constant terms to the opposite side

In order to isolate x, we will move the terms with the variable to the left side and the constant terms to the right side. We will do this by adding 3x to both sides, and adding 10 to both sides: $$15 - 3x +3x = 3x - 10 + 3x$$ $$15 = 6x - 10$$ Now, we will add 10 to both sides to move constant terms to the right side: $$15+10 = 6x - 10 +10$$ $$25 = 6x$$

Solve for x

Now that we have x isolated on one side, we will solve for x by dividing both sides by its coefficient (6): $$\frac{25}{6} = \frac{6x}{6}$$ This simplifies to: $$x = \frac{25}{6}$$ So, the solution to the equation \(15-3x = 3(x-7) +11\) is \(x = \frac{25}{6}\).

Key concepts

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions. It allows you to remove parentheses by distributing a multiplication over addition or subtraction inside the parentheses.
A classic example of this is when we see an expression like a(b + c) = ab + ac.
In the original problem, we have:

• The expression: \(3(x - 7)\)
• We apply the distributive property: \(3 imes x + 3 imes (-7)\)
• Resulting in: \(3x - 21\)

Distributing correctly is important because it ensures that each term within the parentheses is multiplied individually, which sets a foundation for simplifying the equation accurately.
Applying this step helps transform the equation from what appears complex into a more manageable form, clearing the way for the next algebraic steps like combining like terms.

Combining Like Terms

Once an expression is expanded using the distributive property, the next step involves combining like terms to further simplify it. Like terms are terms that have the same variable raised to the same power.
This means you can only combine terms that are similar.
For example, consider 15 - 3x = 3x - 21 + 11. Here, we can:

• Group the variable terms together: \(-3x\) and \(3x\).
• Combine the constant terms: \(-21 + 11\) to get \(-10\).

This results in a simplified equation:

• Left side: \(15 - 3x\)
• Right side: \(3x - 10\)

Effective combining of like terms helps in reducing the equation to fewer terms, making it simpler to manipulate and solve in subsequent steps.

Isolating Variables

Isolating the variable means rewriting the equation to get the variable by itself on one side.
This involves moving all terms containing the variable to one side of the equation, while moving constant numbers to the opposite side.
In the problem, this is achieved by:

• Adding \(3x\) to both sides: this gives \(15 = 6x - 10\).
• Then, adding 10 to both sides to move the constant: \(25 = 6x\).
• Finally, dividing both sides by 6 to solve for \(x\): \(x = \frac{25}{6}\).

The key is to perform operations that simplify and reorganize the equation incrementally until the variable is isolated.
Correctly isolating the variable is crucial because it allows us to directly find its value, providing the solution to the algebraic equation.